Tag - Probability theory

Francesco De Vecchi: Non-commutative Lp-spaces and Grassmann stochastic analysis

12th June, 2023

We introduce a theory of non-commutative Lp spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann Itô processes, Girsanov’s formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables, including a Grassmann analog of the φ24 Euclidean QFT in a bounded region.

Sky Yang Cao: Recent Results on Finite Group Lattice Gauge Theories

20th March, 2023

The rigorous study of spin systems such as the Ising model is currently one of the most active research areas in probability theory. In this talk, I will introduce one particular class of such models, known as lattice gauge theories (LGTs), and go over its origins, motivations, and then some recent results. Along the way, I will also try to highlight some of the key differences between LGTs and the usual spin systems. The general theme is that LGTs are spin systems with topological considerations.

Scott Sheffield: An Introduction to Random Surfaces

28th November, 2022

The theory of 'random surfaces' has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, and mathematical physics. I will give a friendly (I hope) colloquium-level overview of the subject with lots of pictures. Topics will include random planar maps (interpreted as discrete random surfaces), Liouville quantum gravity surfaces, conformal field theory. and the random fractal curves produced from the Schramm-Loewner evolution.  Many of these topics are motivated by physics (statistical physics, string theory, quantum field theory, etc.) but they also have simple mathematical definitions that can be understood without a lot of physics background.

Rafał Kulik: Blocks estimators in Extreme Value Theory

10th November, 2022

Extreme value theory deals with large values and rare events. These large values tend to cluster in case of temporal dependence. This clustering behaviour is widely observed in practice. I will start with a mild introduction to extreme value theory, discussing probabilistic and statistical issues. This part will be accessible to a broader audience.

Then, I will talk about a more specific problem of statistical theory for cluster functionals and rare events. Two types of estimators are of a primary importance: disjoint and sliding blocks estimators. It has been conjectured that sliding blocks estimators are “better” (to be made precise in the talk). We proved in a recent series of papers that this is not the case and in fact both disjoint and sliding blocks estimators are asymptotically equivalent. This part will be aimed at probabilistic and statisticians.

I will conclude with recent directions in extreme value theory, such as extremes in high dimension, extremes of graphs and networks.

Victor Guerassimov: Random walks on groups. An introduction.

10th March, 2022

Geometric methods proved to be useful in the study of some groups. However the geometry of the Cayley graph of a group is rather different from the geometry of classical geometric objects such as homogeneous spaces of Lie groups. The similarity between these two geometries grows as the scale of observation increases. And the asymptototic behavior of them shows surprising similarity. Random walks is an essential tool in studying large-scale geometry of groups. On the other hand it is an interesting object for probabilists since many properties of general stochastic processes are manifested here in a rather simple form. In my talk, I will provide an elementary introduction to this vast area. No special knowledge beyond the usual university mathematics is required.

László Székelyhidi: Morrey’s conjecture

14th February, 2022

Morrey’s conjecture arose from a rather innocent-looking question in 1952: is there a local condition characterizing 'ellipticity' in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first progress was made with V. Sverak’s ingenious counterexample. Nevertheless, the case pertaining to planar maps remains open despite much progress, and has fascinated many through its interesting connections to complex analysis, geometric function theory, harmonic analysis, probability and martingales, differential inclusions and the geometry of matrix space. In the talk, I will give an overview of some of these connections and some of the recent progress.

Robert McCann: Optimal Transportation, Geometry and Dynamics

11th January, 2022

This is a 24-lecture course, with each lecture being around 80 minutes, given by Robert McCann. It gives an introduction to optimal transport.

This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, nonlinear partial differential equations, and machine learning. The basic problem is to find the most efficient structure linking two or more continuous distributions of mass; think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.

Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), geometric inequalities with sharp constants, image processing, optimal decision making, long time asymptotics of dissipative systems, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, fluid mechanics, physics, economics, and geometry. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provide a framework for adapting powerful ideas from Riemannian and Lorentzian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.

Jimmy He: Random walks on finite fields with deterministic jumps

1st March, 2021

Recently, Chatterjee and Diaconis showed that most bijections, if applied between steps of a Markov chain, cause the resulting chain to mix much faster. However, explicit examples of this speedup phenomenon are rare. I will discuss recent work studyingsuch walks on finite fields where the bijection is algebraically defined. This work gives a large collection of examples where this speedup phenomenon occurs. These walks can be seen as a non-linear analogue of the Chung-Diaconis-Graham process, where the bijectionis multiplication by a non-zero element of the finite field. This work is partially joint with Huy Pham and Max Xu.

Various Speakers: Planar random growth and scaling limits

14th September, 2020

An LMS online lecture course in random growth.

Conformal growth models are motivated by some real-world growth processes, and are constructed using conformal maps. We will introduce the one-parameter Hastings–Levitov model, which is used to describe Laplacian growth and allows us to vary between off-lattice versions of many well studied models. Then we investigate the "small particle" scaling limit, which often entails finding a martingale and relating its behaviour to its analogue for the proposed continuum limit.